A three dimensions of semivectorial finite difference beam propagation method (FD-BPM) in cylindrical coordinates is investigated in order to analyze the optical wave propagation in the bent waveguides with the strong lateral confinement which will keep low radiation loss. The three-dimensional (3-D) semivectorial wave propagation equation is programmed in the quick and unconditionally stable procedure by using the alternating-direction implicit method. This version of FD-BPM could well express the polarization characteristics of modal field near the dielectric interfaces and accurately simulate the wave propagation in bent waveguides even at a very small radius (<100\;\mum). Moreover, it is pointed out that the numerical precision of the algorithm is intensively affected by the propagation loss of bent waveguides along with the propagation step length of beam propagation method.
Hua Deng, Guang Hai Jin, Joseph Harari, Jean Pierre Vilcot, and Didier Decoster, "Investigation of 3-D Semivectorial Finite-Difference Beam Propagation Method for Bent Waveguides," J. Lightwave Technol. 16, 915- (1998)